That record normally goes to the Tristan da Cunha islands in the South Atlantic, which are 2,434km from Saint Helena. But they are only the most remote if you consider Tristan da Cunha island itself and Gough Island as part of the same archipelago. The two islands are both inhabited and only 399km apart from each other. Discounting these, the next most remote is Easter Island, which is 2,075km from the nearest other inhabited spot, Pitcairn Island.